Abstract

We introduce the notion of ordered cyclic weakly (ψ,φ,L,A,B)-contractive mappings, and we establish some fixed and common fixed point results for this class of mappings in complete ordered b-metric spaces. Our results extend several known results from the context of ordered metric spaces to the setting of ordered b-metric spaces. They are also cyclic variants of some very recent results in ordered b-metric spaces with even weaker contractive conditions. Some examples support our results and show that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.

MSC:47H10, 54H25.

Keywords

1 Introduction and preliminaries

The Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. One of the interesting generalizations of this basic principle was given by Kirk et al. [1] in 2003 by introducing the following notion of cyclic representation.

for allx∈Aandy∈B, wherek∈[0,1)is a constant. ThenThas a unique fixed pointuandu∈A∩B.

It should be noted that cyclic contractions (unlike Banach-type contractions) need not be continuous, which is an important gain of this approach. Following the work of Kirk et al., several authors proved many fixed point results for cyclic mappings, satisfying various (nonlinear) contractive conditions.

Berinde initiated in [2] the concept of almost contractions and obtained several interesting fixed point theorems. This has been a subject of intense study since then, see, e.g., [3–7]. Some authors used related notions as ‘condition (B)’ (Babu et al. [8]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [9]), and for four maps (Aghajani et al. [10]). See also a note by Pacurar [11]. Here, we recall one of the respective definitions.

A function φ:[0,+∞)→[0,+∞) is called an altering distance function if the following properties hold:

1.

φ is continuous and non-decreasing.

2.

φ(t)=0 if and only if t=0.

So far, many authors have studied fixed point theorems, which are based on altering distance functions.

The concept of a b-metric space was introduced by Bakhtin in [13], and later used by Czerwik in [14, 15]. After that, several interesting results about the existence of fixed points for single-valued and multi-valued operators in b-metric spaces have been obtained (see, e.g., [16–28]). Recently, Hussain and Shah [29] obtained some results on KKM mappings in cone b-metric spaces.

Consistent with [15] and [28], the following definitions and results will be needed in the sequel.

Let X be a (nonempty) set, and let s≥1 be a given real number. A function d:X×X→R+ is a b-metric if for all x,y,z∈X, the following conditions hold:

(b1) d(x,y)=0 iff x=y,

(b2) d(x,y)=d(y,x),

(b3) d(x,z)≤s[d(x,y)+d(y,z)].

In this case, the pair (X,d) is called a b-metric space.

It should be noted that the class of b-metric spaces is effectively larger than the class of metric spaces, since a b-metric is a metric if (and only if) s=1. Here, we present an easy example to show that in general, a b-metric need not necessarily be a metric (see also [[28], p.264]).

Example 1 Let (X,ρ) be a metric space and d(x,y)=(ρ(x,y))p, where p>1 is a real number. Then d is a b-metric with s=2p−1. Condition (b3) follows easily from the convexity of the function f(x)=xp (x>0).

The notions of b-convergent and b-Cauchy sequences, as well as of b-complete b-metric spaces are introduced in an obvious way (see, e.g., [18]).

It should be noted that in general, a b-metric function d(x,y) for s>1 need not be jointly continuous in both variables. The following example (corrected from [22]) illustrates this fact.

Example 2 Let X=N∪{∞}, and let d:X×X→R be defined by

d(m,n)={0,if m=n,|1m−1n|,if one of m,n is even and the other is even or ∞,5,if one of m,n is odd and the other is odd (and m≠n) or ∞,2,otherwise.

Then considering all possible cases, it can be checked that for all m,n,p∈X, we have

d(m,p)≤52(d(m,n)+d(n,p)).

Thus, (X,d) is a b-metric space (with s=5/2). Let xn=2n for each n∈N. Then

d(2n,∞)=12n→0as n→∞,

that is, xn→∞, but d(xn,1)=2↛5=d(∞,1) as n→∞.

Aghajani et al. [16] proved the following simple lemma about the b-convergent sequences.

In particular, ifx=y, thenlimn→∞d(xn,yn)=0. Moreover, for eachz∈X, we have

1sd(x,z)≤lim infn→∞d(xn,z)≤lim supn→∞d(xn,z)≤sd(x,z).

The existence of fixed points for mappings in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings [30], and then by Nieto and Lopez [31]. Afterwards, this area was a field of intensive study of many authors.

Shatanawi and Postolache proved in [32] the following common fixed point results for cyclic contractions in the framework of ordered metric spaces.

there exist0<δ<1and an altering distance functionψsuch that for any two comparable elementsx,y∈Xwithx∈Aandy∈B, we have

ψ(d(fx,gy))≤δψ(max{d(x,y),d(x,fx),d(y,gy),12(d(x,gy)+d(y,fx))});

(c)

forgis continuous, or

(c′) the space(X,⪯,d)is regular.

Thenfandghave a common fixed point.

Here, the ordered metric space (X,⪯,d) is called regular if for any non-decreasing sequence {xn} in X such that xn→x∈X, as n→∞, one has xn⪯x for all n∈N.

By an ordered b-metric space, we mean a triple (X,⪯,d), where (X,⪯) is a partially ordered set, and (X,d) is a b-metric space. Fixed points in such spaces were studied, e.g., by Aghajani et al. [16] and Roshan et al. [27]. In the last mentioned paper, the following common fixed point results for contractions in ordered b-metric spaces were proved.

Let(X,⪯,d)be a complete orderedb-metric space, and letf,g:X→Xbe two weakly increasing mappings. Suppose that there exist two altering distance functionsψ, φand a constantL≥0such that the inequality

ψ(s4d(fx,gy))≤ψ(Ms(x,y))−φ(Ms(x,y))+Lψ(N(x,y))

holds for all comparablex,y∈X, where

Ms(x,y)=max{d(x,y),d(x,fx),d(y,gy),d(x,gy)+d(y,fx)2s}

and

N(x,y)=min{d(y,gy),d(x,gy),d(y,fx)}.

If either [forgis continuous], or the space(X,⪯,d)is regular, thenfandghave a common fixed point.

In this paper, we introduce the notion of ordered cyclic weakly (ψ,φ,L,A,B)-contractions and then derive fixed point and common fixed point theorems for these cyclic contractions in the setup of complete ordered b-metric spaces. Our results extend some fixed point theorems from the framework of ordered metric spaces, in particular Theorem 2. On the other hand, they are cyclic variants of Theorem 3 with even weaker contractive conditions.

We show by examples that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.

2 Common fixed point results

In this section, we introduce the notion of ordered cyclic weakly (ψ,φ,L,A,B)-contractive pair of self-mappings and prove our main results.

Definition 5 Let (X,⪯,d) be an ordered b-metric space, let f,g:X→X be two mappings, and let A and B be nonempty closed subsets of X. The pair (f,g) is called an ordered cyclic weakly (ψ,φ,L,A,B)-contraction if

(1)

X=A∪B is a cyclic representation of X w.r.t. the pair (f,g); that is, fA⊆B and gB⊆A;

(2)

there exist two altering distance functions ψ, φ and a constant L≥0, such that for arbitrary comparable elements x,y∈X with x∈A and y∈B, we have

Let (X,⪯) be a partially ordered set, and let A and B be closed subsets of X with X=A∪B. Let f,g:X→X be two mappings. The pair (f,g) is said to be (A,B)-weakly increasing if fx⪯gfx for all x∈A and gy⪯fgy for all y∈B.

It follows that φ(d(u,gu))=0. Therefore, d(u,gu)=0, and hence gu=u. Similarly, we can show that if u is a fixed point of g, then u is a fixed point of f.

Second part (construction of a sequence by iterative technique).

Let x0∈A, and let x1=fx0. Since fA⊆B, we have x1∈B. Also, let x2=gx1. Since gB⊆A, we have x2∈A. Continuing this process, we can construct a sequence {xn} in X such that x2n+1=fx2n, x2n+2=gx2n+1, x2n∈A and x2n+1∈B. Since f and g are (A,B)-weakly increasing, we have

x1=fx0⪯gfx0=x2=gx1⪯fgx1=x3⪯⋯⪯x2n+1=fx2n⪯gfx2n=x2n+2⪯⋯.

If x2n=x2n+1, for some n∈N, then x2n=fx2n. Thus, x2n is a fixed point of f. By the first part of proof, we conclude that x2n is also a fixed point of g. Similarly, if x2n+1=x2n+2, for some n∈N, then x2n+1=gx2n+1. Thus, x2n+1 is a fixed point of g. By the first part of proof, we conclude that x2n+1 is also a fixed point of f. Therefore, we assume that xn≠xn+1 for all n∈N. Now, we complete the proof in the following steps.

Step 1. We will prove that

limn→∞d(xn,xn+1)=0.

As x2n and x2n+1 are comparable and x2n∈A and x2n+1∈B, by (2.1), we have

By (2.5) and (2.6), we get that {d(xn,xn+1):n∈N} is a non-increasing sequence of positive numbers. Hence, there is r≥0 such that

limn→∞d(xn,xn+1)=r.

Letting n→∞ in (2.5), we get

ψ(r)≤ψ(r)−φ(r),

which implies that φ(r)=0, and hence r=0. So, we have

limn→∞d(xn,xn+1)=0.

(2.7)

Step 2. We will prove that {xn} is a b-Cauchy sequence. Because of (2.7), it is sufficient to show that {x2n} is a b-Cauchy sequence. Suppose on the contrary, i.e., that {x2n} is not a b-Cauchy sequence. Then there exists ε>0, for which we can find two subsequences {x2mi} and {x2ni} of {x2n} such that ni is the smallest index, for which

which implies that φ(lim infi→∞Ms(x2mi,x2ni−1))=0. By (2.15), it follows that lim infi→∞d(x2mi,x2ni)=0, which is in contradiction with (2.8). Hence {xn} is a b-Cauchy sequence in X.

Step 3 (existence of a common fixed point).

As {xn} is a b-Cauchy sequence in X which is a b-complete b-metric space, there exists u∈X such that xn→u as n→∞, and

limn→∞x2n+1=limn→∞fx2n=u.

Now, without loss of generality, we may assume that f is continuous. Using the triangular inequality, we get

d(u,fu)≤sd(u,fx2n)+sd(fx2n,fu).

Letting n→∞, we get

d(u,fu)≤slimn→∞d(u,fx2n)+slimn→∞d(fx2n,fu)=0.

Hence, we have fu=u. Thus, u is a fixed point of f and, since A and B are closed subsets of X, u∈A∩B. By the first part of proof, we conclude that u is also a fixed point of g. □

The assumption of continuity of one of the mappings f or g in Theorem 4 can be replaced by another condition, which is often used in similar situations. Namely, we shall use the notion of a regular ordered b-metric space, which is defined analogously to the case of the standard metric (see the paragraph following Theorem 2).

Theorem 5Let the hypotheses of Theorem 4 be satisfied, except that condition (b) is replaced by the assumption

(b′) the space(X,⪯,d)is regular.

Thenfandghave a common fixed point inX.

Proof Repeating the proof of Theorem 4, we construct an increasing sequence {xn} in X such that xn→u for some u∈X. As A and B are closed subsets of X, we have u∈A∩B. Using the assumption (b′) on X, we have xn⪯u for all n∈N. Now, we show that fu=gu=u. By (2.1), we have

It follows that φ(lim infn→∞Ms(x2n,u))=0, and hence, by (2.20), that d(u,gu)=0. Thus, u is a fixed point of g. On the other hand, similar to the first part of the proof of Theorem 4, we can show that fu=u. Hence, u is a common fixed point of f and g. □

3 Consequences and examples

As consequences, we have the following results.

By putting A=B=X in Theorems 4 and 5, we obtain improvements of the main results (Theorems 5 and 6) of Roshan et al. [27], i.e., of Theorem 3 of the present paper (note that we have s2 instead of s4 in the contractive condition).

there exist0<δ<1, L≥0and an altering distance functionψsuch that for any comparable elementsx,y∈Xwithx∈Aandy∈B, we have

ψ(s2d(fx,fy))≤δψ(Ms(x,y))+Lψ(N(x,y)),

(3.3)

whereMs(x,y)andN(x,y)are given in Corollary 3;

(c)

fis continuous, or

(c′) the space(X,⪯,d)is regular.

Thenfhas a fixed pointu∈A∩B.

Remark 1 (Common) fixed points of the given mappings in Theorems 4 and 5 and Corollaries 3 and 4 need not be unique (see further Example 4). However, it is easy to show that they must be unique in the case that the respective sets of (common) fixed points are well ordered (recall that a subset W of a partially ordered set is said to be well ordered if every two elements of W are comparable).

We illustrate our results with the following two examples.

Example 3 Consider the b-metric space (X,d) given in Example 2, ordered by natural ordering and a mapping f:X→X given as

fn={8n,if n∈N,∞,if n=∞.

If A={n:n∈N}∪{∞} and B={8n:n∈N}∪{∞}, then A∪B is a cyclic representation of X with respect to f. Take ψ:[0,+∞)→[0,+∞) given as ψ(t)=t, δ=5/42 (<1) and L≥0 arbitrary. In order to check the contractive condition (3.3), consider the following cases.

Finally, if x=∞ and y is an odd integer, then d(x,y)=5 and (3.3) trivially holds.

Hence, all the conditions of Corollary 4 are satisfied. Obviously, f has a (unique) fixed point ∞, belonging to A∩B.

We now present an example showing that there are situations where our results can be used to conclude about the existence of (common) fixed points, while some other known results cannot be applied.

Example 4 Let X={0,1,2,3,4} be equipped with the following partial order:

⪯:={(0,0),(1,1),(1,2),(2,2),(3,2),(3,3),(4,2),(4,4)}.

Define a b-metric d:X×X→R+ by

d(x,y)={0,if x=y,(x+y)2,if x≠y.

It is easy to see that (X,d) is a b-complete b-metric space with s=49/25. Set A={0,1,2,3,4} and B={0,2}, and define self-maps f and g by

f=(0123402222),g=(0123402243).

It is easy to see that f and g are (A,B)-weakly increasing mappings with respect to ⪯, and that f and g are continuous. Also, A∪B=X, f(A)⊆B and g(B)⊆A.

Define ψ:[0,∞)→[0,∞) by ψ(t)=t. One can easily check that the pair (f,g) satisfies the requirements of Corollary 1, with any δ and L≥0, as the left-hand side of the contractive condition (3.1) is equal to 0 for all comparable x, y such that x∈A and y∈B. Hence, f and g have a common fixed point. Indeed, 0 and 2 are two common fixed points of f and g. (Note that the ordered set ({0,2},⪯) is not well ordered).

However, take x=1∈A and y=0∈B (which are not comparable). Then

ψ(s2d(f1,g0))=s2(2+0)2=2s>3>3δ+L⋅0=δψ(Ms(1,0))+Lψ(N(1,0)),

where 0<δ<1 and L≥0 are arbitrary, since

Ms(1,0)=max{d(1,0),d(1,2),d(0,0),d(1,0)+d(0,2)2s}=32

and

N(1,0)=min{d(0,0),d(1,0),d(0,2)}=0.

Hence, this result cannot be applied in the context of b-metric spaces without order.

4 Application to existence of solutions of integral equations

Integral equations like (4.1) have been studied in many papers (see, e.g., [22, 33] and the references therein). In this section, we look for a nonnegative solution to (4.1) in X=C([0,T],R).

Consider the integral equation

u(t)=∫0TG(t,s)f(s,u(s))dsfor all t∈[0,T],

(4.1)

where T>0, f:[0,T]×R→R and G:[0,T]×[0,T]→[0,∞) are continuous functions.

Let X=C([0,T]) be the set of real continuous functions on [0,T]. We endow X with the b-metric

D(u,v)=maxt∈[0,T](u(t)−v(t))2for all u,v∈X.

Clearly, (X,D) is a complete b-metric space (with the parameter s′=2). We endow X with the partial order ⪯ given by

x⪯y⟺x(t)≤y(t)for all t∈[0,T].

Clearly, the space (X,⪯,D) is regular.

Let α,β∈X and α0,β0∈R such that

α0≤α(t)≤β(t)≤β0for all t∈[0,T].

(4.2)

Assume that for all t∈[0,T], we have

α(t)≤∫0TG(t,s)f(s,β(s))ds

(4.3)

and

β(t)≥∫0TG(t,s)f(s,α(s))ds.

(4.4)

Let for all s∈[0,T], f(s,⋅) be a decreasing function, that is,

x,y∈R,x≥y⟹f(s,x)≤f(s,y).

(4.5)

Assume that γ>0 is such that

4γ(maxt∈[0,T]∫0TG(t,s)ds)2<1.

(4.6)

Define a mapping T:X→X by

Tu(t)=∫0TG(t,s)f(s,u(s))dsfor all t∈[0,T].

Suppose that for all s∈[0,T] and for all comparable x,y∈X with (x(s)≤β0 and y(s)≥α0) or (x(s)≥α0 and y(s)≤β0),

Now, all the conditions of Corollary 2 (with T=g=f and L=0) hold, and T has a fixed point z in

A1∩A2={u∈C([0,T]):α(t)≤u(t)≤β(t), for all t∈[0,T]}.

That is, z∈A1∩A2 is the solution to (4.1). □

Declarations

Acknowledgements

The authors are highly indebted to the referees of this paper who helped us to improve it in several places. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support. The fourth author is thankful to the Ministry of Education, Science and Technological Development of Serbia.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, King Abdulaziz University

(2)

Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University

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